Subgroups — #20295508

Order 6976.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182030	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄₃₆	C₂	1	3	yes	maximal, complemented
4	#16812661	C₂	1	3	yes	maximal, complemented
5	#16812656	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₁₀₉	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	C₄	1	8	yes	complemented
9	C₂×D₄₃₆	C₂²	1	3	yes	—
10	C₂×C₄×D₁₀₉	C₂²	1	6	yes	—
11	C₂×C₄×D₁₀₉	C₂²	1	3	yes	—
12	C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
13	#13182008	C₂²	1	3	yes	—
14	#13182008	C₂²	1	1	yes	characteristic
15	#13182006	C₂²	1	3	yes	—
16	#13182006	C₂²	1	3	yes	—
17	C₄×Dic₁₀₉	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
20	C₄²×C₁₀₉	C₂²	1	3	yes	—
21	C₂²×D₁₀₉	C₂³	1	3	yes	—
22	Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
25	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
26	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₉	C₂³	1	3	yes	—
28	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
29	D₄₃₆	C₂×C₄	1	12	yes	complemented
30	C₄×D₁₀₉	—	4	6	no	—
31	C₄×D₁₀₉	C₂×C₄	1	12	yes	complemented
32	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
33	C₂×D₁₀₉	—	2	3	no	—
34	C₂×D₁₀₉	—	2	3	no	—
35	C₂×D₁₀₉	C₂²×C₄	1	6	yes	—
36	C₄×C₁₀₉	—	2	3	no	—
37	C₄×C₁₀₉	—	2	1	no	—
38	C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
39	Dic₁₀₉	—	2	3	no	—
40	Dic₁₀₉	—	2	1	no	—
41	Dic₁₀₉	C₂²×C₄	1	2	yes	—
42	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
45	D₁₀₉	—	2	6	no	—
46	C₁₀₉	#520	1	1	yes	characteristic, complemented, Sylow
47	#520	—	109	1	no	maximal, Sylow
48	C₂×#41	—	109	1	no	—
49	C₄×Q₈	—	109	1	no	—
50	C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
51	C₄×D₄	—	109	3	no	—
52	C₄×D₄	—	109	3	no	—
53	#117	—	109	3	no	—
54	C₂×C₄⋊C₄	—	109	3	no	—
55	#41	—	109	8	no	—
56	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
57	C₂×D₄	—	109	3	no	—
58	C₂²×C₄	—	109	3	no	—
59	C₂²×C₄	—	109	6	no	—
60	C₄⋊C₄	—	109	3	no	—
61	C₄⋊C₄	—	109	1	no	—
62	C₄⋊C₄	—	109	3	no	—
63	C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
64	C₂²⋊C₄	—	109	3	no	—
65	C₂²⋊C₄	—	109	3	no	—
66	C₄²	—	109	3	no	—
67	C₄²	C₂×D₁₀₉	1	3	yes	—
68	C₂³	—	109	3	no	—
69	Q₈	C₄×D₁₀₉	1	4	yes	complemented
70	D₄	—	109	12	no	—
71	C₂×C₄	—	109	1	no	—
72	C₂×C₄	—	109	12	no	—
73	C₂×C₄	—	109	1	no	—
74	C₂×C₄	—	109	3	no	—
75	C₂×C₄	—	436	6	no	—
76	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
78	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
79	C₂²	—	109	6	no	—
80	C₂²	—	218	3	no	—
81	C₂²	—	218	3	no	—
82	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	109	2	no	—
86	C₄	—	218	3	no	—
87	C₄	—	218	1	no	—
88	C₄	C₂×C₄×D₁₀₉	1	6	yes	—
89	C₂	—	218	6	no	—
90	C₂	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
91	C₂	#16812745	1	1	yes	characteristic
92	C₂	#16812747	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂⁴
hypercenter	C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	#520
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Q₈×Dic₁₀₉	C₂	semidirect
3. C₄×D₄₃₆	C₂	semidirect
4. #16812661	C₂	semidirect
5. #16812656	C₂	semidirect
6. C₄⋊C₄×D₁₀₉	C₂	semidirect
7. C₄×Q₈×C₁₀₉	C₂	semidirect
8. #13182030	C₄	semidirect
22. Q₈×C₁₀₉	C₂×C₄	semidirect
29. D₄₃₆	C₂×C₄	semidirect
31. C₄×D₁₀₉	C₂×C₄	semidirect
46. C₁₀₉	#520	semidirect
50. C₄×Q₈	D₁₀₉	semidirect
69. Q₈	C₄×D₁₀₉	semidirect